July 05, 2026
A compactly supported RBF is radially symmetric function \(\phi:\mathbb{R}^d\to \mathbb{R}\), with \(\mathrm{supp}\,\phi\subset B(0,1)\) which is positive definite: meaning that for any finite set \(X=\{x_1,\dots x_n\}\), the collocation matrix \(\Phi_X =\bigl(\phi(x_j-x_k)\bigr)_{j,k\le n}\) is strictly positive definite.
In this article, we consider compactly supported RBFs which have have finite global smoothness \(\phi\in C^{\lambda_1}(\mathbb{R}^d)\). Our main object of study is to consider the (higher) smoothness away from the origin: \(\phi\in C^{\lambda_2}\bigl(\mathbb{R}^d \setminus \{0\}\bigr)\) for some \(\lambda_2>\lambda_1\), possibly infinity. When \(\lambda_2\) is finite, we are especially interested in the case that it is sharp (so that \(\phi\notin C^{\lambda_2+\epsilon}\bigl(\mathbb{R}^d \setminus \{0\}\bigr)\) for \(\epsilon>0\)). This functions as a useful secondary parameter for the RBF, which we call the eccentric smoothing parameter.
The advantage of an increased eccentric smoothing parameter was noted in [1], where it was used to obtaining high order convergence rates in high order Sobolev norms for approximation and interpolation with compactly supported RBFs. There are other advantages.
Recently, the compressibility of the matrix \(\Phi_X\), which is full (despite the compact support of \(\phi\)) and becomes ill-conditioned for large or poorly arranged point sets \(X\), has been a favorable outcome of the study of samplets.
For certain RBFs, after a change of basis (represented by the matrix \(\mathcal{T}\)), the collocation matrix \(\Phi_X^{\Sigma}= \mathcal{T}\Phi_X \mathcal{T}^{-1}\) can be rendered sparse with little cost. The ability to compress the matrix relies on the smoothness of the kernel away from the origin.
Specifically, it relies on the asymptotic smoothness of the kernel. This is a concept which has migrated from the theory of compression of \(\mathcal{H}\) matrices [2]. The original condition is very strong condition and compactly supported functions cannot satisfy it. However, a weaker version, namely asymptotic smoothness of finite order, has been considered for positive definite kernels, where it is key to producing wavelet-like bases (i.e., samplets) for kernel spaces in [3], [4]. Indeed, for globally supported RBFs, this type of compression has been used in [5] to produce an effective multiscale algorithm for RBF interpolation with computational cost \(\mathcal{O}(N\log N)\), where \(N=\Xi\), and \(\Xi\) is assumed to be quasi-uniform. This notion of asymptotic smoothness provides sufficient conditions for compressibility of the transformed collocation matrix – we expect that the smoothness away from the origin provided by Proposition 1 will allow compactly supported kernels to be used effectively in this setting.
The integral operator of \(\phi\) plays an important role in kernel approximation, machine learning and statistics [6]–[8], the square root of the operator acting on \(L_2(\sigma)\) is the kernel’s “native space”, the eigenvectors of the operator are fundamental to Mercer and Karhunen-Loève expansions (see [6] [8] for background). The range of the operator on \(L_2(\sigma)\) is the doubling class considered in [9].
When \(\phi\) is compactly supported, convolution with tempered distributions is well defined and continuous, and the action \(\phi*:\mathcal{S}'(\mathbb{R}^d)\to \mathcal{S}'(\mathbb{R}^d):f\mapsto \phi*f\) can be described by way of the Fourier transform (it is a multiplier). However, if \(\lambda_2\) is finite, its symbol \(a:(x,\xi) \mapsto \hat{\phi}(\xi)\), considered as a map on \(\mathbb{R}^d\times \mathbb{R}^d\), does not belong to any of the Hörmander symbol classes \[S_{\rho,\delta}^{\tau}(\mathbb{R}^d)= \bigl\{ a\in C^{\infty}(\mathbb{R}^d\times \mathbb{R}^d) \mid (\forall \alpha, \beta)\, (\exists C_{\alpha,\beta})\; |D_x^{\beta}D_{\xi}^{\alpha} a(x,\xi)| \le C_{\alpha,\beta} \llbracket \xi\rrbracket^{\tau - \rho|\alpha|+\delta||\beta|} \bigr\}.\] with \(0\le \delta<\rho\le 1\) (see [10] for background). Here we use \(\llbracket \xi\rrbracket := (1+|\xi|^2)^{1/2}\). In short, the integral operator fails to be a Hörmander class pseudodifferential operator. This fact can be verified in the following two ways:
if \(j\) is sufficiently large, then \(x\mapsto |x|^{2j} \phi(x)\) is in \(C^{\lambda_2}(\mathbb{R}^d)\), but not smoother. Thus for any \(j>(\lambda_2-\lambda_1)/2\), \(|\Delta_{\xi}^{j}\widehat{\phi}(\xi)|\) cannot decay like \(|\xi|^{-\mu}\) for any \(\mu> \lambda_1+d+1\).
if \(\phi*\) were a pseudodifferential operator with symbol in \(S_{\rho,\delta}^{\tau}(\mathbb{R}^d)\) with \(\rho>0\) and \(\delta<1\), then it would have the pseudolocal property: for any compactly supported distribution \(T\), \(\mathrm{sing\, supp} (\phi* T) \subset \mathrm{sing\, supp} (T)\) (see [10]). This fails because \[\mathrm{sing\, supp}(\phi* \delta) = \mathrm{sing\, supp}( \phi) = \{0\} \cup \partial B(0,1) \supsetneq \{0\} =\mathrm{sing\, supp} (\delta).\]
As a consequence, understanding the mapping properties of \(\phi*\) on Sobolev spaces \(W_p^s(\mathbb{R}^d)\) when \(p\neq 2\) (or other classes like Besov and Triebel-Lizorkin spaces), or of the map \(f\mapsto \int_{\mathbb{R}^d} f(y) \phi(x-y) \mathrm{d}\sigma(y)\) on \(L_p(\sigma)\) when \(\sigma\) is a finite Borel measure remains difficult. Both reduce to understanding how \(\phi*\) acts on tempered distributions. This stands in sharp contrast to other well-known (globally supported) RBFs having analytically similar Fourier transforms, such as Matérn kernels and surface splines, which induce pseudodifferential operators [11].
In this brief article, we demonstrate how to modify, simply, standard constructions of compactly supported RBFs to produce compactly supported RBFs with similar analytic properties but which are smooth away from origin. This is Proposition 1. This yields kernels with asymptotic smoothness of finite order. The infinitely smooth kernels constructed using Proposition 1 have integral operators which are properly supported pseudo-differential operators with symbols in the Hörmander class. This is demonstrated in Proposition 4. As a consequence, these integral operators have prescribed smoothing properties.
There are a number of constructions of compactly supported positive definite functions: [12]–[16]. See also [17] for a recent overview.
In this article, we focus on a general framework which includes Wendland’s original compactly supported RBFS introduced in [15] and further discussed in [18] as well as “generalized Wendland" functions, introduced and considered in [19]. We exploit the analytic properties collected in [1], which can also be readily found in the literature (specifically in [19] and [18]).
A continuous, positive definite RBF \(\phi:\mathbb{R}^d\to \mathbb{R}\) is polyhomogeneous if there is a univariate polynomial \(p\) so that \[\label{Wend} \phi(x) = \begin{cases}p(|x|), &|x|< 1\\ 0&|x|\ge 1.\end{cases}\tag{1}\] By splitting \(p\) into even and odd parts and expanding the odd part in monomials, we have \[p(r)= \sum_{\ell=0}^M c_{\ell} r^{{\lambda_1} + 1+2\ell} + q(r^2),\] for some univariate polynomial \(q\), integers \(0\le \lambda_1\le M\) and coefficients \(c_{\ell}\) with \(c_0\neq 0\). In this case, \(\lambda_1\) is the global smoothness of \(\phi\) in that \(\phi\in C^{{\lambda_1},1}(\mathbb{R}^d)\) (the partial derivatives of order \({\lambda_1}\) are Lipschitz); but \(\phi\notin C^{{\lambda_1}+1}(\mathbb{R}^d)\).
Define \(\mathbb{R}^d_* := \mathbb{R}^d\setminus\{0\}\). A consequence of (1 ) is that there is a positive integer \(\lambda_2\), the eccentric smoothness, for which \(\phi|_{\mathbb{R}^d_*}\in C^{{\lambda_2}}(\mathbb{R}^d_*)\setminus C^{{\lambda_2}+1}(\mathbb{R}^d_*)\). By [18], one has \(\lambda_2>\lambda_1+\lfloor d/2\rfloor\). This brings us to a general form for compactly supported RBFs we will use throughout this note:
Definition 1. Let \(\phi:\mathbb{R}^d\to \mathbb{R}\) be a compactly supported RBF satisfying (1 ) with global smoothness \(\lambda_1\), and eccentric smoothness \(\lambda_2>\lambda_1 + \lfloor d/2\rfloor\). Suppose furthermore that there are constants \(0<C_1\le C_2<\infty\) so that \[\label{fourier} C_1 \llbracket \xi \rrbracket^{-2t}\le \widehat{\phi}(\xi) \le C_2 \llbracket \xi \rrbracket^{-2t}\tag{2}\] with \(t := ({\lambda_1}+d+1)/2\).
The original compactly supported RBFs of Wendland given in [15] and further discussed in [18] satisfy the requirements of Definition 1 with \(\lambda_2 = \lambda_1 +\lfloor\frac{d-1}{2}\rfloor\). Constructions with larger values of \(\lambda_2\) can be found in [1] and [19]. In particular, the class of generalized Wendland functions \(\phi_{\mu, \alpha}\) defined in [19] satisfy (1 ) in [19], for certain integer values of \(\mu\) and \(\alpha\). They satisfy (2 ) in [19] for fairly general choices of parameters \(\mu, \alpha\).
It follows from the definition that \({\lambda_1}\) determines the order of the RBF in the sense that the reproducing kernel Hilbert space (the so-called native space) it generates via the Moore-Aronszajn theorem is \(W_2^t(\mathbb{R}^d)\). It is shown in [1] that by considering longer series, i.e., for greater polynomial degrees \(\deg(p)\), in (1 ), it is possible to increase the smoothness \({\lambda_2}\) near the support boundary boundary without changing the kernel’s core analytic characteristics: the global smoothness \({\lambda_1}\), the Fourier decay parameter \(t=({\lambda_1}+d+1)/2\), or the native space \(W_2^t(\mathbb{R}^d)\).
There are other ways to increase \(\lambda_2\) without sacrificing the core analytic characteristics – we call this eccentric smoothing and consider it and its benefits below.
The construction below requires an RBF \(\phi\) satisfying Definition 1 and another, smoother, compactly supported RBF \(\tau\) which is otherwise arbitrary (it can be any of the ones considered in [12]–[16] or any other method, provided the support and smoothness conditions are satisfied).
Proposition 1. Suppose \(\phi\) is a positive definite RBF satisfying Definition 1 with global smoothness \(\lambda_1\) and eccentric smoothness \({\lambda_2}\ge 2t=\lambda_1+1+d\). Suppose furthermore that \(\tau\) is a positive definite RBF with support in \(B(0,1)\):
If \(\tau \in C^k(\mathbb{R}^d)\) with \(k\ge \max(2t,\lambda_2)\), and \(\int_{\mathbb{R}^d} \tau(x) \mathrm{d}x >0\), then \(\tilde{\phi} = \tau \phi\) lies in \(C^{{\lambda_1},1}(\mathbb{R}^d)\), with \(\tilde{ \phi}|_{\mathbb{R}_*^d} \in C^{k}(\mathbb{R}_*^d)\);
If \(\tau\) has the form given in Definition 1, with global smoothness \(\kappa_1\ge\lambda_1\) and eccentric smoothness \(\kappa_2\), then \(\tilde{\phi} = \tau \phi\) lies in \(C^{{\lambda_1},1}(\mathbb{R}^d)\), with \(\tilde{ \phi}|_{\mathbb{R}_*^d} \in C^{\kappa_2+{\lambda_2}+1}(\mathbb{R}_*^d)\).
In each case, the Fourier transform of \(\tilde{\phi}\) satisfies \[\tilde{C}_1\llbracket \xi\rrbracket^{-2t} \le \widehat{(\tau\phi)}(\xi) \le \tilde{C}_2\llbracket \xi\rrbracket^{-2t}\] for some \(0 < C_1\le C_2\), and \(\tilde{\phi}\) is positive definite.
Case 2 shows that the class of functions satisfying Definition 1 is closed under pointwise multiplication. Consequently, computing an eccentrically smoothed RBF of this type is not harder than computing the original function \(\phi\). Indeed, for a Wendland or generalized Wendland function \(\phi\), one may take the pointwise power \(\phi^n\); from the point of view of RBF interpolation on a set \(X\), this is simply a matter of taking \(n\)th powers of the matrix entries of \(\Phi_X\).
Proof. The fact that \(\tau \times \phi \in C^{{\lambda_1},1}(\mathbb{R}^d)\) follows from the product rule.
The eccentric smoothness in case 1 follows from the product rule, as well.
In case 2, \(\tau\) and \(\phi\) both vanish to high degree on \(\partial B(0,1)\): \(D^{\alpha} \bigl(\tau\phi\bigr)(x) = \sum_{\beta\le \alpha} C_{\beta,\alpha} D^{\beta} \tau(x) D^{\alpha-\beta} \phi(x)\) by Leibniz’s rule; if \(|x|=1\), any such term sum is nonzero only if both \(|\beta|>\kappa_2\) and \(|\alpha-\beta|>\lambda_2\); thus \(D^{\alpha}\tilde{\phi}(x) =0\) for all \(|x|=1\) and all \(|\alpha|\le \kappa_2+\lambda_2+1\), and the product is in \(C^{\kappa_2+{\lambda_2}+1}\) in a neighborhood of the sphere, and thus on \(\mathbb{R}_*^d\).
The Fourier transform of \(\tilde{\phi}\) is given by the convolution \(\widehat{\tau \phi}(\xi) = [\hat{\tau}*\hat{\phi}](\xi)\). This guarantees strict positivity of the Fourier transform and therefore positive definiteness of \(\tilde{\phi}\). Furthermore, Peetre’s inequality guarantees the convolution estimate \[\int \llbracket\zeta\rrbracket^{-N_1} \llbracket\xi-\zeta\rrbracket^{-N_2} \mathrm{d}\zeta \le C \llbracket \xi \rrbracket^{-\min(N_1,N_2)}\] (see also [20]), which we can use to get the desired bounds on the Fourier transform of \(\tilde{\phi}\).
In each case, \(\widehat{\tau}(\xi) \le C \llbracket \xi\rrbracket^{-2t}\). (In case 1, \(\tau \in C_c^k(\mathbb{R}^d)\) guarantees \(\widehat{\tau}(\xi) \le C \llbracket \xi\rrbracket^{-k}\) for some \(C\), while in case 2 it follows from Definition 1 and the fact that \(\kappa_1\ge \lambda_1\).) An application of the above convolution inequality with \(N_2= 2t = N_1\), ensures the upper bound.
For the lower bound, positivity and continuity of \(\hat{\tau}\) ensure the \(\hat{\tau}(z)>c>0\) on a neighborhood \(B(0,\delta)\); thus we have \[[\hat{\tau}*\hat{\phi}](\xi) = \int \hat{\phi}(\zeta) \hat{\tau}(\xi-\zeta)\mathrm{d}\zeta \ge \int_{|\xi-\zeta|<\delta} \hat{\phi}(\zeta) \hat{\tau}(\xi-\zeta) \mathrm{d}\zeta \ge c \int_{|\xi-\zeta|<\delta} \hat{\phi}(\zeta)\mathrm{d}\zeta\] Using the lower bound \(\hat{\phi}(\zeta) \ge C_1\llbracket \zeta \rrbracket^{-2t}\) and noting there is a \(\delta\)-dependent constant \(C'\) so that for \(\zeta\in B(\xi,\delta)\), \(\llbracket \zeta \rrbracket \le C' \llbracket \xi \rrbracket\), we have \[[\hat{\tau}*\hat{\phi}](\xi) \ge c \int_{|\xi-\zeta|<\delta} \hat{\phi}(\zeta) \mathrm{d}\zeta \ge c C_1 (C')^{-2t} \mathrm{vol}(B(\xi,\delta)) \llbracket \xi \rrbracket^{-2t} .\] and the result follows with \(\tilde{C}_1 = c (C')^{-2t} \mathrm{vol}(B(\xi,\delta))>0.\) ◻
Let \(\varrho\) be a positive integer. A kernel \(\kappa:\mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}\) is “\(\varrho\) - asymptotically smooth” in the sense of [3] if for all multi-integers \(\alpha, \beta\) for which \(|\alpha|,|\beta|\le \varrho\), there exists a constant \(c_{\alpha,\beta}\) so that \[\label{eq:asympsmooth} |D_x^{\alpha}D_y^{\beta} \kappa(x,y) | \le c_{\alpha,\beta} |x-y|^{-(|\alpha|+|\beta|)}.\tag{3}\] Kernels which are \(\varrho\) - asymptotically smooth have collocation matrices which can be nicely compressed by using samplets of order \(\varrho\), which we discuss below.
A samplet basis of order \(\varrho\), \[\Sigma = \Phi_0 \cup \bigcup_{j=0}^J \Sigma_j\subset \mathrm{span}_{\xi\in\Xi} \delta_{\xi},\] is a collection of finitely supported measures constructed by a multiresolution analysis (MRA) which is a basis for \(\mathrm{span}\{ \delta_{\xi}\mid \xi\in\Xi\}\), and for which every samplet \(\sigma_{j,k} \in \Sigma_j\) annihilates \(\mathcal{P}_{\varrho-1}(\mathbb{R}^N)\) (i.e., has \(\varrho-1\) vanishing moments). Indeed, \(\Sigma\) is an orthonormal basis in the induced \(\ell_2(\Xi)\) inner product given by \(\|\sum_{\xi\in \Xi} a_{\xi} \delta_{\xi}\|^2 = \sum_{\xi\in\Xi} |a_{\xi}|^2\).
Associated to each samplet \(\sigma_{j,k}\) is a cluster \(\nu_{j,k} \subset \Xi\) with \(\mathop{\mathrm{supp}}(\sigma_{j,k})\subset \nu_{j,k}\), The clusters (and therefore the supports) have the property that for every \(j,k\) and \(j',k'\) with \(j'\ge j\), \(\nu_{j',k'}\subset \nu_{j,k}\) or \(\mathrm{conv}(\nu_{j,k}) \cap \mathrm{conv}(\nu_{j',k'}) =\emptyset\). Indeed, the clusters are the nodes of a binary tree structure with \(\nu_0 = \Xi\) at the root, and were every where every non-leaf node \(\nu_{j,k}\) has precisely two children \(\nu_{j+1,k_1}\) and \(\nu_{j+1,k_2}\) which satisfy \(\nu_{j,k} =\nu_{j+1,k_1} \cup \nu_{j+1,k_2}\). See [4] for the construction of the cluster three, the MRA, as well as presentation of desirable properties of the samplets.
The standard collocation matrix \(K_{\Xi} =\bigl(\kappa(\xi,\zeta)\bigr)_{\xi,\zeta\in\Xi} = (\langle \kappa, \delta_{\xi}\otimes \delta_{\zeta}\rangle)_{\xi,\zeta\in\Xi}\) can be easily transformed via a change of basis to \(K_{\Xi}^{\Sigma} = \langle \kappa,\sigma\otimes\sigma' \rangle_{\sigma,\sigma' \in \Sigma}\) (this change of basis is unitary, by the orthonormality of the samplets). For a \(\varrho\)-asymptotically smooth kernel, the collocation matrix in samplet coordinates \(K_{\Xi}^{\Sigma}\) can be nicely compressed by [3], to produce a sparse matrix. For a user defined \(\eta>0\), the matrix \(K_{\eta}\) is obtained by truncating \(K_{\Xi}^{\Sigma}\) by setting \(\bigl({K}_{\eta}\bigr)_{\sigma,\sigma'} = 0\) if the clusters \(\nu\) and \(\nu'\) of \(\sigma\) and \(\sigma'\) (respectively) are distant in the sense that \(\mathrm{dist}(\nu,\nu') \ge \eta \max(\mathrm{diam}(\nu),\mathrm{diam}(\nu'))\). For this setup, there is a constant \(c\) so that \[\|K_{\Xi}^{\Sigma}-K_{\eta}\|_{F} \le c_{\text{samp}}\eta^{-2\varrho} N \log(N), \qquad \text{where }N =\#\Xi.\] This is precisely [3]; furthermore, [3] shows that for quasi-uniform sets \(\Xi\), a relative error of \(\frac{\|K_{\Xi}^{\Sigma}-K_{\eta}\|_{F}}{\|K_{\Xi}^{\Sigma}\|_{F}}\le c\eta^{-2\varrho}\) with compressed matrix \(K_{\eta}\) having \(\mathcal{O}(N \log N)\) nonzero entries.
Naturally, for such estimates to be useful, it makes sense to assume \(\eta>1\), since the case \(\eta\le 1\) in the estimates above can already be accomplished by setting \(K_{\eta}=0\). (Note, for instance, that the uniform bound on the diagonal entries \(\sup_{\xi\in \mathbb{R}^d} \kappa(\xi,\xi) \le C\) guarantees that the entries of \(K_{\Xi}^{\Sigma}\) are bounded, and so \(\|K_{\Xi}^{\Sigma}\|_{F} = \|K_{\Xi}\|_F \le C N\) ).
For translation invariant kernels like compactly supported RBFs, where \(\kappa(x,y) = \phi(x-y)\), the above condition reduces to the following: for all \(|\alpha|\le 2\varrho\), there is \(c_{\alpha}\) so that for all \(x\neq 0\), \[|D_x^{\alpha} \phi(x) | \le c_{\alpha} |x|^{-|\alpha|}.\] Naturally this means \(\phi |_{\mathbb{R}_*^d} \in C^{2\varrho}(\mathbb{R}_*^d)\) – in short, the asymptotic smoothness of a compactly supported RBF is limited by its smoothness at the frontier \(\partial B(0,1)\). The originally considered RBFs satisfying Definition 1 are at most asymptotically smooth of order \(\lambda_2/2\).
On the other hand, as we show below, eccentric smoothing a compactly supported kernel provides greater asymptotic smoothness. It follows that compressibility improves for eccentrically smoothed RBFs.
Since \(\lambda_2>\lambda_1+1\), one could simply take a higher order compactly supported RBF (one with greater global smoothness, a more rapidly decaying Fourier transform and a smaller native space), but this comes at a cost to the condition number of the matrix \(\Phi_{\Xi}\) (as well as its representation in the samplet basis \(\Phi_{\Xi}^{\Sigma}\)). In particular, [18] ensures that the smallest eigenvalue of \(\Phi_{\Xi}\) behaves like \(\lambda_{\min} (\Phi_{\Xi}) \ge c_{\text{eig}} q_{\Xi}^{2t-d} = c_{\text{eig}} q_{\Xi}^{\lambda_1+1}\), where \(t = (\lambda_1+1+d)/2\) and \(q_{\Xi}=\min_{\xi\in\Xi}\min_{ \zeta\in \Xi\setminus\{\xi\}} |\xi-\zeta|\). This leads to a theoretically bounded condition number on the order \(q_{\Xi}^{-2t}\) for point sets exhibiting the bound \(N=\#\Xi \sim q_{\Xi}^{-d}\) (this is a consequence of quasi-uniformity).
Thus one virtue of eccentric smoothing of the compactly supported kernel is that it improves the compressibility of the collocation matrix while preserving its conditioning.
Any compactly supported kernel of the form (1 ) with \(\phi |_{\mathbb{R}_*^N} \in C^{2\varrho}(\mathbb{R}_*^N)\) is \(\varrho\)-asymptotically smooth. In particular, we have the following.
Corollary 2. If \(\tilde{\phi}\) is the eccentrically smoothed RBF considered in case 1 of Proposition 1 then \(\tilde{\phi}\) is \(k/2\) asymptotically smooth.
Proof. The fact that \(\tilde{\phi}|_{\mathbb{R}_*^d} \in C^{k}\), ensures that there is \(C_{\phi}\) so that for any \(|\alpha|\le k\) and for any \(|x|\ge 1/2\), \[|D^{\alpha}\tilde{\phi}(x)| \le C_{\phi} |x|^{-|\alpha|}\] (this uses the fact that \(\tilde{\phi}\) is supported in the unit ball).
To obtain bounds near to the origin, note that homogeneity implies that, for any \(b\in \mathbb{R}\) and \(\alpha\in \mathbb{Z}_+^d\), the estimate \(\bigl|D^{\alpha} |x|^{b}\bigr| \le \gamma_{b,\alpha} |x|^{b-|\alpha|}\) holds for some constant \(\gamma_{b,\alpha}\) and for all \(x\neq 0\). Consequently, for any multi-index \(\alpha\) there is \(C_{\alpha}\) so that if \(|x|\le 1\) then \[|D^{\alpha}\phi(x)| \le \sum_{\ell=0}^M |c_{\ell}| \gamma_{\lambda_1+1+2\ell,\alpha} |x|^{\lambda_1+1+2\ell-|\alpha|} +\bigl|D^{\alpha} q(|x|^2)\bigr| \le C_{\alpha} |x|^{-|\alpha|}.\] Multiplying \(\phi\) by \(\tau\), which is \(C^k\) gives, by Leibniz’s rule, that \[|D^{\alpha}\tilde{\phi}(x)|\le \sum_{\beta\le \alpha} \begin{pmatrix} \alpha\\ \beta\end{pmatrix} \|D^{\beta} \tau\|_{\infty} C_{\alpha-\beta} |x|^{-|\alpha-\beta|} \le C|x|^{-|\alpha|}\] provided \(|\alpha|\le k\). ◻
We can do better than this if we choose \(\tau\) to also be a generalized Wendland function, as given in Definition 1; in particular, by requiring \(\tau\) to be polyhomogeneous on its support.
Corollary 3. If \(\tilde{\phi} = \tau\phi\) is a eccentrically smoothed RBF considered in case 2 of Proposition 1, then \(\tilde{\phi}\) is \((\kappa+\lambda_2+1)/2\) asymptotically smooth.
Proof. Since the product of polyhomogeneous functions is polyhomogeneous, we have for \(|x|\le 1\), \[\tilde{\phi}(x) = \sum_{\ell =0}^{\tilde{M}} a_{\ell} |x|^{2a+1+2\ell} + \tilde{q}(|x|^2).\] Consequently, for all \(0<|x|<1\), \(|D^{\alpha} \tilde{\phi} (x) | \le C |x|^{-|\alpha|}\) for some constant \(C\) depending on \(\alpha\), and the coefficients \(a_{\ell}\) and \(\tilde{q}\) (in particular, this holds for all \(|\alpha|\ge 0\)). On the other hand, \(\tilde{\phi} \in C^{\kappa_2+\lambda_2+1}(\mathbb{R}_*^d)\) and vanishes outside of \(B(0,1)\), so the estimate \(|D^{\alpha} \tilde{\phi} (x) | \le C |x|^{-|\alpha|}\) extends to \(\mathbb{R}_*^d\) as long as \(|\alpha|\le \kappa_2+\lambda_2+1\). ◻
One application of this is the following. Suppose, one is interested in solving a linear system with the compressed matrix \(K_{\Xi}^{\Sigma} x\approx K_{\eta} \tilde{x} =f\) in order to obtain an approximate solution. In order to solve the system with the compressed matrix, one needs that \(K_{\eta}\) is regular (and by construction symmetric), i.e., that its lowest eigenvalue is bounded below. This follows by classical perturbation of eigenvalue results from \[\|K_{\Xi}^{\Sigma}-K_{\eta}\|_{2} \le c_{\text{samp}}\eta^{-2\varrho} q_{\Xi}^{-d} d | \log(q_{\Xi})| \le c_{\text{samp}}\eta^{-2\varrho} q_{\Xi}^{-d} d q^{-1}_{\Xi} \le \frac{1}{2}c_{\text{eig}} q_{\Xi}^{\lambda_1+1} {\le} \frac{1}{2}\lambda_{\min} (\Phi_{\Xi}).\] Now, using Corollary 3 with \(2\varrho=\kappa+\lambda_2+1\), we obtain that \(K_{\eta}\) is positive definite for \(\eta\gtrsim q_{\Xi}^{-\frac{\lambda_1+d+2}{\kappa+\lambda_2+1}}\). As \(\eta\) determines the sparsity of \(K_{\eta}\), the inequality ensuring positive definiteness might also be used to determine the eccentric smoothing parameter \(\kappa\) if \(\eta\) is considered fixed by enforcing enough compression to make matrix operations feasible.
It follows that there exists for any \(\lambda_2>2t\) a compactly supported RBF \(\phi\) with native space \(W_2^t(\mathbb{R}^d)\) and boundary smoothness \(\phi|_{\mathbb{R}_*^d}\in C^{\lambda_2}(\mathbb{R}_*^d)\). Notably, by considering the standard mollifier \(\varphi\in C_c^{\infty}(\mathbb{R}^d)\) with support in \(B(0,1/2)\) the iterated function \[\tau = \varphi*\varphi\] has Fourier transform \(\widehat{\tau}(\xi) = |\widehat{\varphi}(\xi)|^2 \ge 0\) which is analytic and thus has only isolated zeros on \(\mathbb{R}^d\). Consequently, \(\tau\) is an infinitely smooth RBF with support in \(B(0,1)\). By using this function in Proposition 1, we can obtain compactly supported RBFs with Sobolev native space \(W_2^t(\mathbb{R}^d)\) and singleton singular support: \(\mathrm{sing \,supp}(\phi )= \{0\}\). See for instance [21] for an exposition on how to compute these kernels numerically as no closed-form expression seems to be available. See also the recent paper [22] for a Mercer type approach towards smooth, compactly supported kernels, which are not radial.
Proposition 4. Let \(\phi\) be a compactly supported RBF of the form (1 ), and let \(\tilde{\phi}= \phi*\tau\) be a compactly supported RBF obtained from Proposition 1 with \(\tau \in C^{\infty}(\mathbb{R}^d)\). Then the integral operator \(\tilde{\phi}\!*\) is pseudodifferential with symbol in \(S_{1,0}^{-2t}(\mathbb{R}^d)\).
Proof. Fix \(\alpha\) It suffices to show that \(|D^{\alpha}\mathcal{F}\tilde{\phi}(\xi) |\le C\llbracket \xi\rrbracket^{-(2t+|\alpha|)}\) for all \(\xi\). Since \(\mathcal{F}\tilde{\phi}(\xi)\) is continuous, it suffices to demonstrate this inequality for \(|\xi|\) sufficiently large (we will describe this in the next paragraph).
This follows because \(\Phi:\mathbb{R}^d\to \mathbb{R}\), defined by \(\Phi(x) =\sum_{\ell=0}^M c_\ell |x|^{\lambda_1+1+2\ell} +q(|x|^2)\) for all \(x\) (not just for \(|x|\le 1\)), as a combination of homogeneous functions, satisfies the following: there exist constants \(\Gamma_{\alpha}\) and \(R_{\alpha}>0\) so that for any \(|\xi|\ge R_\alpha\), the inequality \[\label{polyharmonic} |D_{\xi}^{\alpha} \widehat{\Phi}(\xi)| \le \Gamma_{\alpha} \llbracket \xi\rrbracket^{-2t-|\alpha|}\tag{4}\] holds. Set \(|\xi|>4R_\alpha\).
Since \(\tau\) is supported on \(B(0,1)\), the functions \(\phi\tau\) and \(\Phi \tau\) are identical. Thus \(\tilde{\phi}=\Phi \tau\) has distributional Fourier transform \(\mathcal{F} \tilde{\phi}= [\widehat{\Phi}* \widehat{\tau}]\) which satisfies for \(\xi\neq 0\) \[D^{\alpha}\mathcal{F}\tilde{\phi}(\xi) = [\widehat{\tau}* D^{\alpha} \widehat{\Phi}](\xi) = \langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),\widehat{ \tau}\rangle\] Using a smooth cut-off \(\upsilon\), supported in \(B(0,2R_{\alpha})\) and equaling one on \(B(0,R_{\alpha})\), we express the above pairing as \[D^{\alpha}\mathcal{F}\tilde{\phi}(\xi) = \langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),G\rangle + \langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),B\rangle\] where \(B=\widehat{ \tau}\upsilon(\xi-\cdot)\) and \(G = \widehat{ \tau}\bigl(1-\upsilon(\xi-\cdot)\bigr)\) are Schwartz functions, having supports \(\mathrm{supp}(B)\subset B(\xi,2R_\alpha)\) and \(\mathrm{supp}(G)\subset \mathbb{R}^d\setminus B(\xi,R_\alpha)\).
To treat \(|\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),B\rangle|\), we note that because \(D^{\alpha} \widehat{\Phi}(\xi - \cdot)\) is a tempered distribution, \[|\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),B\rangle| \le C_{\Phi}\sum_{|\alpha|\le N,|\beta|\le M} \rho_{\alpha,\beta}(B)\] for some constant \(C_{\Phi}\) and for semi-norms \(\rho_{\gamma,\beta} (B)= \sup_{\xi \in \mathbb{R}^d}|x^{\gamma}D^{\beta}B(\xi)|\). Since \(\widehat{\tau}\in \mathcal{S}(\mathbb{R}^d)\), the fact that \[\sup_{\zeta \in B(\xi,2R_{\alpha})} |\zeta^{\gamma} D^{\beta}\widehat{\tau}(\zeta)| \le C_{\gamma,\beta} \min_{\zeta\in B(\xi,2R_{\alpha})}|\zeta|^{-M}\] for each \(\gamma,\beta\) implies that for any \(M>0\) there is \(C_M\) so that \[\sum_{|\alpha|\le N,|\beta|\le M} \rho_{\alpha,\beta}(B) \le C_M (1 +|\xi| -2R_{\alpha})^{-M}.\] Thus \[|\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),B\rangle| \le C_{\Phi} \sum_{|\alpha|\le N,|\beta|\le M} \rho_{\alpha,\beta}(B) \le C_{\Phi} C_{M} (1+|\xi|-2R_\alpha)^{-M}\le C \llbracket \xi\rrbracket^{-M} .\] for any \(M\) – in particular, for \(M>2t+|\alpha|\).
To treat \(\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),G\rangle\), we note that the pairing is represented by an integral, since \(\widehat{\Phi}\) is regular on \(\mathbb{R}_*^d\). Thus, for \(M>2t+|\alpha|+d\), we have \[\begin{align} |\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),G\rangle | &=& \left|\int_{|\xi-\zeta|>R_\alpha} D^{\alpha} \widehat{\Phi}(\xi - \zeta) \widehat{\tau}(\zeta) \bigl(1-\upsilon(\xi-\zeta)\bigr) \mathrm{d}\zeta\right|\\ &\le& \Gamma_{\alpha}C_M \int_{|\xi-\zeta|>R_{\alpha}} \llbracket \xi -\zeta\rrbracket^{-2t-|\alpha|} \llbracket\zeta\rrbracket^{-M} \mathrm{d}\zeta. \end{align}\] Here we have used (4 ) and the fact that \(\tau\) is a Schwartz function. To finish the estimate, we use Peetre’s inequality \(\llbracket\xi -\zeta\rrbracket^{-2t-|\alpha|} \le 2^{2t+|\alpha|} \llbracket \xi \rrbracket^{-2t-|\alpha|} \llbracket \zeta \rrbracket^{2t+|\alpha|}\) to get \[|\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),G\rangle | \le C \llbracket \xi \rrbracket^{-2t-|\alpha|} .\] Adding this to the bound for \(|\langle D^{\alpha} \widehat{\Phi}(\xi - \cdot),B\rangle|\) gives \[|D^{\alpha}\mathcal{F}\tilde{\phi}(\xi) |= \bigl|D^{\alpha}[\widehat{\Phi}* \widehat{\tau}](\xi)\bigr|\le C \llbracket \xi \rrbracket^{-2t-|\alpha|}\] and the result follows. ◻
We now consider mapping properties of the induced integral operator – for this we consider the scale of Besov spaces \(B_{p,q}^s(\mathbb{R}^d)\). What we present here can easily be modified to handle Triebel-Lizorkin spaces \(F_{p,q}^s(\mathbb{R}^d)\); standard Sobolev spaces \(W_p^s(\mathbb{R}^d)\) for \(1<p<\infty\) are either Besov (for fractional spaces) or Triebel-Lizorkin (for integer spaces); the cases \(p=1,\infty\) in the Triebel-Lizorkin scale are more difficult.
To define Besov spaces, we begin by fixing a spectral partition of unity. Let \(\psi:\mathbb{R}^d\to [0,\infty)\) be a bandlimited (Schwartz) test function with annular support \(\mathrm{supp}(\widehat{\psi})\subset B(0,2)\setminus B(0,1/2)\) and which satisfies \(\sum_{k=-\infty}^{\infty} \widehat{\psi}(2^k \xi) =1\) for all \(\xi\in\mathbb{R}^{d}\setminus\{0\}\). Define \[\label{fourier95multiplier} \psi_k:=2^{dk} \psi(2^{k} \cdot)\tag{5}\] to be a dyadic dilation of \(\psi\). Define \(\omega \in \mathcal{S}(\mathbb{R}^d)\) by \(\widehat{\omega}= 1-\sum_{k=1}^{\infty}\widehat{ \psi}(2^{-k}\cdot)\)
Definition 2. For \(s\in \mathbb{R}\) \(p,q\in[1,\infty]\), the Besov space \(B_{p,q}^{s}(\mathbb{R}^d)\) consists of tempered distributions, \(f\), for which \(f*\omega\in L_p\), each \(\psi_k*f\in L_p\) and for which the norm \[\| f\|_{B_{p,q}^{s}(\mathbb{R}^d)}:= \|f*\omega\|_{L_p(\mathbb{R}^d)}+ \Bigl\| k\mapsto 2^{ks} \|\psi_k*f\|_{L_p(\mathbb{R}^d)} \Bigr\|_{\ell_q(\mathbb{N})}\] is finite.
Corollary 5. Let \(\tilde{\phi}\) be the eccentrically smoothed, compactly supported RBF of Proposition 4. Then for any \(p,q\in [1,\infty]\) and any \(s\in \mathbb{R}\), the map \[\tilde{\phi}*: B_{p,q}^s(\mathbb{R}^d)\to B_{p,q}^{s+2t}(\mathbb{R}^d): f\mapsto \tilde{\phi}*f\] is bounded.
Proof. Since \(\tilde{\phi}*\) is proper, its composition with other pseudodifferential operators is well-defined. In particular, for the Bessel potential operator, \(J^{2t}:B_{p,q}^s(\mathbb{R}^d) \mapsto B_{p,q}^{s-2t}(\mathbb{R}^d)\) defined by \((J^s f)^{\wedge} (\xi) = (1+|\xi|^2)^{s/2}\widehat{f}(\xi)\) (see [23] or [24]) we have that \(J^{2t}\circ \tilde{\phi}*\) is a pseudodifferential operator in \(S_{1,0}^{0}(\mathbb{R}^d).\)
The result follows as a direct consequence of regularity properties of Hörmander class pseudodifferential operators; see for instance [24]: since the composition \(J^{2t}\circ \tilde{\phi}*\) is a pseudodifferential operator in \(S_{1,0}^0\), the map \(J^{2t}\circ \tilde{\phi}*:B_{p,q}^{s}(\mathbb{R}^d)\to B_{p,q}^s(\mathbb{R}^d)\) is continuous, the composition \(\phi* =J^{-2t} \circ J^{2t}\circ \tilde{\phi}*: B_{p,q}^{s}(\mathbb{R}^d)\to B_{p,q}^{s+2t}(\mathbb{R}^d)\) is continuous as well. ◻
Let \(\mu\) be a finite, signed Borel measure on \(\mathbb{R}^d\), with Jordan decomposition \(\mu= \mu^+-\mu^-\), total variation measure \(|\mu| = \mu^+ + \mu^-\) and total variation norm \(\|\mu\|_{TV} = |\mu|(\mathbb{R}^d)\). Then Young’s theorem gives \(\|\mu*f\|_p\le \|\mu\|_{TV} \|f\|_p\). This implies \(\|\mu *\psi_k\|_p\le 2^{kd/p'}\|\psi\|_p \|\mu\|_{TV}\), where \(p'\) is the conjugate exponent to \(p\), i.e., \(1/p+1/p'=1\). Thus \(\mu\in B_{p,\infty}^{-d/p'}(\mathbb{R}^d)\) and \[\label{finite95measure95besov} \|\mu\|_{B_{p,\infty}^{-d/p'} (\mathbb{R}^d)}\le C_{\psi} \|\mu\|_{TV}\tag{6}\] holds with a constant \(C_{\psi}\) depending on the spectral partition of unity induced by \(\psi\).
Corollary 6. Let \(\tilde{\phi}\) be the eccentrically smoothed, compactly supported RBF of Proposition 4. If \(\mu\) is a finite, signed Borel measure on \(\mathbb{R}^d\) then the integral operator \[\mathcal{L}_{\tilde{\phi}}:L_p(\mu) \to S'(\mathbb{R}^d): f\mapsto \int_{\mathbb{R}^d} f(y) \tilde{\phi}(\cdot-y)\mathrm{d}\mu(y)\] has range in \(B_{p,\infty}^{2t-d/p'} (\mathbb{R}^d)\) and \(\mathcal{L}_{\tilde{\phi}}: L_p(\mu) \to B_{p,\infty}^{2t-d/p'} (\mathbb{R}^d)\) is continuous.
Proof. Since \(\mu\in B_{p,\infty}^{-d/p'}(\mathbb{R}^d)\), where \(p' = (1-1/p)^{-1}\), it follows that \(L_p(\mu)\subset B_{p,\infty}^{-d/p'}(\mathbb{R}^d)\) as well, in the sense that \(f\times \mu: U\mapsto \int_{U} f(x) \mathrm{d}\mu(x)\) is a finite measure and \(\|f\times \mu\|_{TV} \le \|f\|_{L_p(\mu)} \|\mu\|_{TV}^{1/p'}\). Indeed, for any \(g\in C_0(\mathbb{R}^d)\), \[\left| \int_{\mathbb{R}^d} g(x) f(x) \mathrm{d}\mu(x) \right| \le \|g\|_{\sup} \int_{\mathbb{R}^d} |f(x)|\, \mathrm{d}|\mu|(x) \le \|g\|_{\sup} \|f\|_{L_p(\mu)} \bigl( |\mu|(\mathbb{R}^d) \bigr)^{1/p'}.\] Consequently, \(\|f\times \mu\|_{B_{p,\infty}^{-d/p'}(\mathbb{R}^d)} \le C_{\psi}\|f\times \mu\|_{TV} \le C_{\psi} \|f\|_{L_p(\mu)} \|\mu\|_{TV}^{1/p'}\) holds.
It follows that \(f\times \mu\) is a tempered distribution and that \(\tilde{\phi}* (f\times \mu) = \int_{\mathbb{R}^d} f(y) \tilde{\phi}(x-y) \mathrm{d}\mu(y)\). By Corollary 5, \[\left\| \int_{\mathbb{R}^d} f(y) \tilde{\phi}(\cdot-y) \mathrm{d}\mu(y) \right\|_{B_{p,\infty}^{2t-d/p'} (\mathbb{R}^d)} \le C_{\psi} \|f\times \mu\|_{B_{p,\infty}^{-d/p'}(\mathbb{R}^d)} \le {C} \|f\|_{L_p(\mu)} \|\mu\|_{TV}^{1/p'},\] where \(C\) incorporates the constants from (6 ) and the operator norm \(\|\tilde{\phi}*\|_{B_{p,\infty}^{-d/p'}(\mathbb{R}^d) \to B_{p,\infty}^{2t-d/p'}(\mathbb{R}^d)}\). ◻
The above situation may be improved for specific \(\mu\) (for instance if \(\mu\) is Hausdorff measure of some compact set); in that case the range may be in a smaller Besov space with smoothness index greater than \(2t-d/p'\) – [11]. Secondly, if \(\mu\) is a Borel measure with infinite variation, then the above holds for \(p=1\) (since in that case, \(f\times \mu\) is a finite measure – this is the case for Lebesgue measure).